Improved Accuracy for Nanomechanical Measurements with Force-Curve-Based AFM Techniques

One of the most popular ways to obtain nanomechanical information by atomic force microscopy is single-point measurements that track the force applied on the sample versus the cantilever’s Z-piezo position. These measurements are generally known as force spectroscopy or force curves. A variety of contact mechanics models can be fitted to the force curves in order to extract useful properties of the sample, such as adhesion and stiffness (modulus). These measurements have been available since the earliest days of atomic force microscopy and are extensively used, but despite this, they present a number of challenges such as slow acquisition speed for mapping and the need to calibrate multiple parameters related to the atomic force microscope (AFM) cantilever.

In order to address the speed problem, FASTForce Volume™ has been integrated on Bruker Dimension Icon^®, BioScope Resolve^®, Dimension FastScan^® and MultiMode^® AFM systems in order to improve force spectroscopy measurements. Leveraging the traditional force curves, FASTForce Volume™ makes the same measurement but at much higher ramp rates, thus decreasing acquisition time tenfold. For instance, a force curve map of 128x128 pixels earlier took 30 minutes, but now it takes just about 3 minutes. The same maps of height, adhesion, and modulus in real time are still carried out, but at a faster speed, which allows for higher measurement throughput.

PeakForce Tapping^® is another, even faster method for force-curve mapping. PeakForce Tapping^® was first introduced in 2009 and since then, it has become a popular tool for AFM measurements. It is a simple, easy-to-understand technique that requires only minimal setup beyond laser alignment and cantilever installation, and it is relevant to virtually any material and any surface. When combined with ScanAsyst^®, which optimizes the key operating parameters, PeakForce Tapping^® is one of the easiest atomic force microscopy methods to set up.

PeakForce Tapping^® is a non-resonant technique based on force curves performed with direct force control at ultralow forces. This method provides several well-known advantages such as minimal lateral forces, minimal wear on the tip, and high resolution. The PeakForce Tapping^® platform has been successfully integrated with capabilities to define other material properties. These comprise of methods for characterizing both mechanical properties (PeakForce QNM^®) and electrical properties (PeakForce KPFM™, PeakForce TUNA™, PeakForce sMIM™).

PeakForce QNM^® builds on PeakForce Tapping^® by examining the force curves in real time to extract relevant material properties, such as deformation, indentation, adhesion, and the DMTModulus (modulus by fitting to the DMT model). Similar to other AFM methods, PeakForce QNM^® has been subject to challenges when it comes to calibrating the system and modeling the behavior of the sample under load by the AFM tip, which can restrict the repeatability and accuracy of nanomechanical measurements. Bruker has developed a series of major advancements in both software and hardware that provide significant improvements in the productivity and performance of PeakForce QNM^® and FASTForce Volume™.

Probe Manufacturing and the Calibration of Spring Constant and Tip Shape

Probe tip shape has been one of the major challenges in obtaining quantitative AFM nanomechanical measurements. Tip size and geometry plays an important role in the contact mechanics used for modeling tip-sample interactions. The traditional Hertzian model (or Hertzian-based DMT, which includes some adhesion beyond the tip-sample contact) is one of the most popular contact mechanics models used to model the interactions between AFM tip and sample. In this model, there is a direct, well-known association between the load exerted on the sample (F) by a spherical probe with radius (R):

Equation 1.

(where E* is the reduced modulus, and d is the deformation of the sample). The tip radius as well as the tip shape appears clearly in this equation, the latter as the exponent 3/2 on the deformation (d), which relies on the tip geometry. Both are highly variable in commercial AFM tips that are normally used for imaging purposes.

Bruker now offers pre-calibrated probes with rounded and well-defined tips that provide confidence in the tip shape and diameter. The pre-calibrated probes are available in a wide range of spring constants from 0.25 N/m to 200 N/m to access measurements on many different samples. Their rounded tips are separately measured through SEM to guarantee a nominal radius of 30 nm, which easily transitions into a cone with a half angle of 25 degrees. This affords a controlled contact area for various indentation depths up to 100 nm.

For work on cells, pre-calibrated probes with a larger 65 nm radius are also available. A SEM image of a sample probe is shown in Figure 1. This consistent geometry is important especially for measurements on heterogeneous samples, where a constant load will result in different indentations on the different components based on the sample’s material properties. With this well-defined geometry, all of a sample’s components will experience a predictable contact area, even at different indentation depths.

Figure 1. New, pre-calibrated probes with a controlled tip geometry are now available from Bruker. Calibrated parameters of the cantilever such as tip radius, spring constant, resonance frequency, and quality factor are conveniently stored in a QR code.

Using a laser doppler vibrometer (LDV), the spring constant of each individual probe is also pre-calibrated to offer the most accurate measurement available for this critical cantilever parameter and to prevent the user from having to do this step. Therefore, photodiode sensitivity is the only remaining unknown parameter in Equation 1 and can be calibrated through a thermal tune, or alternatively by performing a force curve on a stiff sample, for example sapphire. By scanning the QR code on each cantilever box, the pre-calibrated parameters, such as the resonance frequency, spring constant, tip radius, tip half-angle, and quality factor, are easily read into the system. The user merely has to choose which probe is being employed in the experiment and all the parameters are suitably populated into the system calibration and control parameters file.

Z Calibration

Z piezo calibration has been one of the major challenges of implementing PeakForce QNM^®. PeakForce Tapping^® (and PeakForce QNM^®) uses a sinusoidal Z motion where the Z piezo position is defined as:

Equation 2.
Z = A sin(2πƒ + φ)

Where, the amplitude (A) and phase (ϕ) can differ with frequency and rely on both precise configuration and the system. New parameters (PFT Amplitude Sens for amplitude and Sync Distance QNM for phase) have been introduced to control these parameters. These new parameters can be calibrated with a few clicks of the mouse. Figure 2 shows the dialog used to perform this calibration.

On a stiff sample such as sapphire, a user-specified number of force curves are gathered and analyzed to calibrate the deflection sensitivity. After measuring the deflection sensitivity, the phase and amplitude parameters are calibrated using the bottom panel. A set of force curves are performed at the frequency of the PeakForce Tapping measurement, and the required PFT Amplitude Sensitivity and QNM Sync Distance are automatically calculated by the software. These parameters can be measured at multiple frequencies and stored for convenient use later on. By using the Z sensor in the Icon scanner, the PFT Amplitude can also be calibrated automatically after engage.

Figure 2. New software dialog for convenient, accurate calibration of the deflection sensitivity, as well as the frequency and phase of the sinusoidal Z motion in PeakForce Tapping.

Improved Modeling in Data Analysis

The PeakForce QNM software now includes new tip shape and contact mechanics models for detailed modeling capabilities to cover a host of materials and tip-sample interactions. The Derjaguin-Muller-Toporov (DMT) model is used by PeakForce QNM imaging for real-time analysis during acquisition. Therefore, the resulting modulus maps are labeled “DMTModulus”. This model is suitable for a wide range of materials, accounting for a small amount of adhesion outside of the contact but other models could be more appropriate for a given sample, for example soft materials with considerable adhesion (e.g., JKR), or where a simpler method (e.g., Hertz or Sneddon) that does not require adhesion to be sufficient.

Many different models are now provided for off-line analysis, such as JKR, Hertz, and DMT for parabolic or spherical tips. The Sneddon model is also provided for cone-shaped and pyramidal tips, while a cone-sphere model is provided for tips that start with a spherical apex and transition to a conical shape.

Besides increased variety, other recently implemented software improvements are also available. Numerous algorithms are now available to measure the adhesion, as well as varying methods to fit the baseline. A new PeakForce Capture™ capability offers a means to save a force curve for each pixel in the image. Just like Force Volume, PeakForce Capture data files share the same data structure and can be analyzed in the same view, making it easy to compare the results from PeakForce Tapping with those from Force Volume (using the same analysis tools). This novel comprehensive analysis package is provided in a user-friendly GUI for full flexibility and customization in fitting the PeakForce QNM data on a wide range of materials, and with a range of probes.

Figure 3. PeakForce QNM imaging of a tri-polymer blend of polystyrene (PS), polyethylene (PE), and polypropylene (PP). The PeakForce QNM values for images collected with five different cantilevers are compared with time-temperature superposed DMA values. An error of 5% is included in the DMA values.

Improved Accuracy

Using the above-described enhancements, major improvement in the PeakForce QNM method for quantitative measurements has been seen across a wide range of samples. A PeakForce QNM image of a three-polymer blend of polypropylene (PP), polyethyslene (PE), and polystyrene (PS) is shown in Figure 3. The figure looks like a “snowman” where the background is PP, the dark head is the PE component, and the bright body is PS. This sample was measured with a set of five Bruker RTESPA-300-30 probes. The probes were among the ones illustrated above with LDV pre-calibrated spring constants and 30 nm rounded tips. No reference sample was needed – the only calibration required was afforded by the QR code on the box (Figure 1) and the guided calibration dialog (Figure 2).

The chart shown on the bottom of Figure 3 gives a comparison of the DMT modulus (red bars) as compared with the corresponding modulus determined on a dynamic mechanical analyzer (DMA) in the blue bars. The PeakForce QNM data is basically an average of the data obtained from the five cantilevers. It must be noted that the DMA values have been time-temperature superposed to the high frequency of 2 kHz for suitable comparison with the AFM data.

The moduli for the PS and PP match well with their DMA counterparts and are seen to be well within the error of the measurement. When compared to the DMA PE value, the PE modulus is low. One plausible explanation is that the adhesion on the PE is higher than on the other two components, which may possibly complicate the measurement and subsequent modeling employed for this material. Another possibility is that the processing of the blend could have affected the PE modulus. To sum up, the AFM measurement accurately captures the trend between the three materials and also offers excellent quantitative measurements on the PS and PP – all “out of the box” without using a reference sample.

Bridging over a Large Frequency Range

PeakForce QNM measurements can now be made over a range of frequencies from 125 Hz to 2000 Hz. This allows frequency-dependent measurements, which are specifically interesting for viscoelastic materials that can have considerable frequency dependence. Moreover, FASTForce Volume (FFV) measurements also cover the low end of the spectrum from 1 Hz to more than 100 Hz. All in all, AFM-based force-curve measurements can now bridge the large frequency range between 1 Hz and 2000 Hz.

Figure 4. Force curve mapping of a blend of polystyrene (PS) and low-density polyethylene (LDPE) at a range of ramp rates. Force volume measurements are conducted at a frequency of 0.5 Hz to 10 Hz. PeakForce QNM samples the higher frequency range of 500Hz to 2 kHz. Note that contrast of LDPE starts out as a dark purple at low frequency and changes to a reddish contrast at higher frequencies, corresponding to an increase in modulus for this viscoelastic material. In contrast, the PS is uniformly pink throughout all of the measurements, indicating no change in modulus with frequency as expected. The plot on the right shows the DMT Modulus of the PS (blue diamonds) and the LDPE (red squares) over more than three orders of magnitude of ramp rate.

Shown in Figure 4 is an example of such a bridging experiment on a blend of LDPE and PS. The same area of the sample was imaged using a range of frequencies in force volume and then switched over to PeakForce QNM in order to access the higher frequencies. The modulus (fit to a DMT model) as a function of frequency is therefore plotted for both samples, with the LDPE modulus in red and the PS modulus in blue. The PS modulus stays constant, while the LDPE modulus increases with frequency, emphasizing the LDPE’s frequency dependence and the PS’ frequency independence in this range.

Figure 5a shows another example of a bridging experiment, where the modulus (as determined by a DMT fit) of polyvinyl chloride has been calculated in the identical area from 1 Hz on the bottom all the way to 2 kHz on the top. All the images are displayed on the identical Z scale. The data obtained in 125 Hz through 2 kHz were collected through PeakForce QNM, while the data obtained at 1 Hz, 10 Hz, and 61 Hz were collected in force volume. Figure 5b shows the modulus vs. frequency plot, revealing a slow increase in modulus over this frequency range.

When the frequency increases above 500 Hz, there is improved discrimination of the PVC structure. This material contains particulates and additives that become integrated into the sample during the process of synthesis and molding, and therefore these patches of particles and soft additives become more apparent with high frequency.

Figure 5b shows the modulus plot that was produced by fitting both the retract curve (points in red) and the extend curves (points in blue) or approach; Figures 5c (approach) and Figure 5d (retract) show the fits of individual curves. The DMT model does not consider viscoelastic deformation, which results in different modulus values based on fitting the approach and the curve’s retract portions.

Figure 5. (a) Force volume and PeakForce QNM images of polyvinyl chloride (PVC). Image size 10 μm. Force volume measurements are conducted at a frequency of 1 Hz to 61 Hz while PeakForce QNM measurements are conducted at a frequency of 12 5 Hz to 2 kHz. (b) a plot of modulus (via DMT fit) vs. frequency as fit to the retract portion of the force curve (red points) or approach/extend portion of the force curve (blue points).(c) Sample approach/extend curve at 10 Hz with fit, Er~10 MPa. (d) Sample retract curve with fit, Er~23 MPa.

A word of caution should be considered when comparing these AFM force curve experiments to bulk viscoelastic measurements. Some AFM-based mechanical property measurements, such as force modulation and contact resonance, are likely to use small perturbations to the applied force, and therefore a linear response can be assumed during analysis. On the other hand, tapping-based and force-curve-based methods that employ larger amplitudes make and break contact during the motion cycle and are, hence, not operating in a linear response regime.

Moreover, the frequency experienced by the sample in these measurements is not just the ramp rate or drive frequency parameter that is set by the user. In addition to that frequency, the sample also experiences higher order harmonics owing to motion of the tip during the interaction. These caveats should be kept in mind when making any direct comparison with other frequency based measurements, for example dynamic mechanical analysis (DMA). It is hoped that the AFM community will revisit this topic in the near future as nanomechanical measurements on viscoelastic samples become more accurate and quantitative.

Limits of Uncertainty

Despite all the considerable software and hardware improvements, AFM-based methods are inherently restricted by the uncertainty of system parameters, such as deflection sensitivity, spring constant¸ and tip radius. In order to better comprehend the various contributions and their magnitude to error in force volume measurements of modulus, an error analysis was performed.

As illustrated in Equation 1, the well-known Hertzian-based DMT model relates F (the load exerted by a spherical probe on a surface) to the probe radius (R), reduced modulus (E*), and amount of deformation (d=Z-D) in the sample through the following relationship:

Equation 3.

F=K_c *D where K_c is the cantilever spring constant, and where D=S_DV is the cantilever deflection, and V is the measured deflection voltage, and S_D is the deflection sensitivity.

Thus, the variance formula estimates the error in the reduced modulus, resulting in the following equation:

Equation 4.

In order to estimate the error in modulus, the following (two sigma) errors were assumed in the other parameters: δR ~15%, δK_c ranging from ~6%, 8%, 10%, and 16% for softest to stiffest spring constant, δSd ~5%, δV ~1% and δZ~1%.

Figure 6 shows a plot of the estimated error in modulus by force volume measured for a set of four Bruker cantilevers with constant force (ScanAsyst-Air-HPI-30 [SAA-HPI-30] at k~0.25 N/m; RTESPA150-30 at k~5 N/m; RTESPA300-30 at k~40 N/m; and RTESPA525-30 at k~200 N/m). This analysis was performed with the assumption that a constant force is applied on the sample for each of the cantilevers.

Figure 6. Error analysis of modulus in force volume measurements conducted at constant force for a series of cantilevers. SAA-HPI-30cantilever measurements conducted at force of 2 nN; RTESPA-150-30at 20 nN; RTESPA-300-30 at 100 nN; RTESPA-525-30 at 400 nN.

A baseline error of approximately 12% is revealed by the plot, with error differing considerably based on the cantilever combined with the sample modulus being probed. It is well known that the cantilever stiffness should match the contact stiffness and only then the cantilever can appropriately sample the material. In case the cantilever is too stiff relative to the sample, then only the properties of the cantilever will be measured. If the cantilever is too soft or compliant relative to the sample, there will be inadequate deformation into the sample to determine its properties.

As shown in Figure 6, the incorrect lever for a specific sample can considerably increase the error in the modulus measurement. For instance, the softest lever (SAA-HPI-30) is suitable for samples with a modulus of a few MPa, but produces a major error in modulus of nearly 50% if used to determine sample with modulus of 100 MPa. Similar observations exist for all the levers.

On the whole, the error in the high modulus limit is dominated by error in Z piezo position and deflection sensitivity (Sd), while the error in the low modulus limit is dominated by errors in the cantilever spring constant and tip radius. For modulus measurements at constant deformation, the error increases for soft samples owing to baseline deflection noise (data not shown). Shown in Figure 6 are the limitations on accuracy of the modulus measurement that can be achieved with force volume.

The error somewhat increases for PeakForce QNM measurement, where there is more error in knowing the Z position. Here, the amount of Z position error relies on contact time and error in Sync Distance QNM. For PeakForce QNM with SyncDistance QNM error ~0.12% and contact time of 10%, the predicted modulus errors increase by approximately 3% when compared to force volume error, leading to a baseline error in modulus measurement with PeakForce QNM of approximately 15%.

Guidelines for Implementation

The most vital experimental parameter a user needs to set in force spectroscopy measurements occurs before any experiment even begins, and that is the selection of the spring constant. As described above, selecting a lever that is too soft will result in too much compliance in the lever, poor sampling of the sample, and therefore a large error in the modulus; and choosing a too stiff cantilever results in the same problem through not enough compliance in the lever.

Figure 6 acts as a guideline for spring constant selection depending on the modulus of the sample. A soft lever such as a ScanAsyst-Air-HPI-30, or even a RTESPA-150-30, is suitable for soft materials under 10 MPa. The RTESPA-150-30 functions well in the intermediate range of 10 MPa – 200 MPa. The RTESPA-300-30 with a 40 N/m spring constant covers the vital range of moduli from 200 MPa to approximately 5 GPa. As described above, measurements on extremely soft samples are not practical with the stiffer probes because of increased baseline noise. These modulus recommendations for different cantilever types are summarized in Table 1.

Table 1. Probe recommendations based on estimated sample modulus (E).

Another critical parameter to optimize is the indentation into the sample on each force curve. It is recommended to have an indentation of 1-3 nm into the sample, which can be managed by the trigger threshold parameter. Less than 1 nm indentation means that the probe may not be adequately penetrating into the sample. Higher indentations (>100 nm) are likely to contaminate the probe needlessly and can also access the part of the tip where the geometry of the tip is not controlled. Enough data points should be acquired in the contact part of the curve so that the contact mechanics can get a reasonable fit. This can be achieved by either decreasing the ramp size or increasing the number of sample points for each curve in force volume. In PeakForce QNM, it can be done by decreasing the PFT frequency, decreasing the PFT amplitude, or by increasing the PeakForce Set point.

Conclusions

FASTForce Volume and PeakForce QNM provide easy and highly productive force-curve-based platforms for nanomechanical measurements. In the last several years, major enhancements have been made to improve the accuracy, flexibility, and productivity of these vital tools to determine the properties of materials, such as adhesion and modulus. Bruker AFMs now include the latest, easy-to-follow calibration steps coupled with well-defined, pre-calibrated AFM tips for better accuracy in measurements.

In addition, new software capabilities in analysis and modeling expand the utility of the measurement to a boarder range of samples. Lastly, probing a range of properties as a function of frequency can now be achieved over a wide range, thus broadening horizons to fully investigate the properties of viscoelastic materials.

This information has been sourced, reviewed and adapted from materials provided by Bruker Nano Surfaces.

For more information on this source, please visit Bruker Nano Surfaces.